r - The Hong Kong Polytechnic University

United States.) The available site data include velocity logs, boring logs and site-specific material curves
(except the Kiknet site).
Shear Wave Velocity Profile
The median velocity profiles are estimated based on available geophysical measurements and used as the
baseline geotechnical model. Variability in the velocity profiles is estimated either from data (if multiple
measurements at the same site are available) or based on the empirical model of Toro (1997). The Toro model is
based on 176 velocity profiles from the Savannah River site, and is a statistical model that can be used to
estimate the standard deviation and correlation in velocity profiles for both generic (broad geographic region)
and site-specific conditions. In current analyses, the alternative velocity profiles are obtained by adding and
subtracting to the baseline profile 3 times the depth-dependent site-specific standard deviation. The 3
factor is used because the theoretical optimal three-point representation of a normal distribution involves
sampling the distribution at the mean (μ) and μ ± 3σ
, and then providing the samples with weights of 2/3 and
1/6 (twice). Those sample points and weights preserve the first (mean), second (variance), and fourth central
moments of the underlying distribution (Rosenblueth, 1975; Ching et al., 2006).
Modulus Reduction and Damping Curves
The modulus reduction and damping curves are used to model the nonlinear soil behavior for different depth
ranges. The median (baseline) curves used in current analyses are based on either material-specific laboratory
testing, inference of in-situ material property from vertical array data or Darendeli (2001) statistical model.
Variability in material curves is considered by adding and subtracting to the baseline curves 3 times the
strain-dependent standard deviation. This standard deviation is taken either from Darendeli (2001) or estimated
from in-situ data.
RESULTS AND DISCUSSIONS
Ground response analyses are performed for the vertical array sites using various nonlinear ground response
analysis codes with both baseline (median) and alternative (taking into account the variability in material
properties) geotechnical models.
To evaluate the model-to-model variability for the ground surface results for period T, the median estimate
ln( S a
( T ))
is first evaluated from the five nonlinear model predictions. Model variability, σ m , is then calculated
from the variance as follows:
∑
⎡ln ( S ) ( ) 2
a( T) − ln S ( )
,
a
T ⎤
⎣
pre i
⎦
2
i
σ
m( T) = Var( Sa( T)
) =
(1)
pre
N −1
where N = number of predictions (five).
The response variability due to material uncertainty is assessed using DEEPSOIL predictions only. To calculate
the standard deviation due to velocity, ground motions are predicted based on two non-baseline velocity profiles
(mean + 3 standard deviation velocities and mean - 3 standard deviation velocities). The standard
deviation of the ground motions due to the variability in velocity (denoted σ v ) is estimated according to the
FOSM, method (Baker and Cornell, 2003; Melchers, 1999), as follows:
3
2
σ
v
= ∑ wi(ln( Sa( T) i
−ln( Sa( T))
(2)
where
i=
1
Sa( T ) = S ( T )
Sa( T ) = S ( T )
1 a Vs
: μ
2 a Vs: μ+
3σVs
Sa( T ) = S ( T )
3 a Vs: μ−
3σV s
3
a
= ∑ i a i
i=
1
ln( S ( T)) w ln( S ( T) )
w1 = 2/3; w2 = w3
= 1/6
The standard deviation due to the variability in material curves (denoted σ G ) is estimated similarly to σ v .
In Figure 1, uncertainties in predictions due to different sources of variability for all four vertical array sites are
(3)
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